#331919

order	512
nilpotency class	3
isoclinism	isoclinic class of #331791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

660 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,7)(4,6)(5,8)(9,10,13,17)(11,18,14,15)(12,20,21,23)(16,19,24,22)(25,26,29,30)(27,33,35,31)(28,32,37,38)(34,36,40,39);(1,3)(2,5)(4,8)(6,7)(9,11,13,14)(10,15,17,18)(12,19,21,22)(16,23,24,20)(25,27,29,35)(26,31,30,33)(28,36,37,39)(32,34,38,40);(10,15)(11,14)(16,23)(17,18)(19,22)(20,24)(25,28)(26,32)(27,34)(29,37)(30,38)(31,39)(33,36)(35,40);(2,6)(3,8)(9,12,13,21)(10,16,17,24)(11,19,14,22)(15,23,18,20)(25,26,29,30)(27,33,35,31)(28,32,37,38)(34,36,40,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,f^a=fh,c^b=cg,d^b=deh,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 239115);